Torque balance at a line of contact

Kerins and Boiteux (Physica A 117 (1983) 575) were the first to apply Noetherʹs theorem to the van der Waals theory of non-uniform fluids. In particular, for a three-phase line of contact, they showed that the translation invariance of the variational integral for the excess free energy implies a balance of forces at the three-phase line. In this paper we consider the implications of the rotation invariance of the variational integral for the excess free energy. Intuitively, one would expect this invariance to lead to an equation of torque balance, and this is indeed the case—the total moment of the forces around the line of contact is zero. In the course of the calculation it will be necessary to find an expression for the surface of tension for a model with a multi-component density, which is a simple extension of earlier work by Fisher and Wortis (Phys. Rev. 29 (1984) 6252). At the same time, we extend to multi-component densities those authors’ results for the Tolman length and the equimolar surface, making contact with a more general calculation by Groenewold and Bedeaux (Physica A 214 (1995) 356).